## Abstract

Fire-related hazards and incidents are a very common phenomenon that affects human society heavily, so many organisations, over a long period of time, have made efforts to mitigate fires and the caused damages. It is widely acknowledged that an evaluation of fire protection performance is critical for such efforts. In this paper, we propose a new data envelopment analysis-based approach for fire protection efficiency analysis in the United States at the state level. For this purpose, the fire protection system is generalised as an innovative two-stage network process, in which the fire hazard defence subsystem in the first stage is followed by a fire incident fighting subsystem in the second stage. Further, both intermediate outputs and final outputs are all undesirable measures. The fire protection expenditure, a kind of shared resources, is modelled under managerial disposability since it can be intentionally used to reduce fires and fire damages. Based on an empirical study of data from 2010 to 2014, we find that: (1) Texas, California, New York, Florida, and Illinois are the top five states prone to fire incidents. (2) The United States as a whole has an average fire protection efficiency of 0.590, implying relatively low fire protection performances at the state level. (3) Wyoming, Vermont, and Rhode Island are the top three most efficient states, whereas Iowa, Arkansas, and Pennsylvania are ranked the least efficient. Further, (4) the fire incident fighting efficiency is more likely to be higher than the fire hazard defence efficiency for most states. (5) Looking at larger scale by grouping all states into eight areas, the Far West has the highest fire protection efficiency, while Southeast and Plains areas have the lowest efficiency scores. Interestingly, (6) the results show also that a 1% increase in the fire protection expenditure by states per capita will result in a 2.6893% improvement in fire protection efficiency.

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## References

Banker RD, Charnes A, Cooper WW (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Manag Sci 30(9):1078–1092

Bian Y, Liang N, Xu H (2015) Efficiency evaluation of Chinese regional industrial systems with undesirable factors using a two-stage slacks-based measure approach. J Clean Prod 87:348–356

Charnes A, Cooper WW (1962) Programming with linear fractional functionals. Nav Res Log (NRL) 9(3–4):181–186

Charnes A, Cooper WW, Rhodes E (1978) Measuring the efficiency of decision making units. Eur J Oper Res 2(6):429–444

Chen Y, Cook WD, Li N, Zhu J (2009) Additive efficiency decomposition in two-stage DEA. Eur J Oper Res 196(3):1170–1176

Chen Y, Du J, Sherman HD, Zhu J (2010) DEA model with shared resources and efficiency decomposition. Eur J Oper Res 207(1):339–349

Chung YH, Färe R, Grosskopf S (1997) Productivity and undesirable outputs: a directional distance function approach. J Environ Manag 51(3):229–240

Cook WD, Liang L, Zhu J (2010) Measuring performance of two-stage network structures by DEA: a review and future perspective. Omega 38(6):423–430

Cooper WW, Park KS, Pastor JT (1999) RAM: a range adjusted measure of inefficiency for use with additive models, and relations to other models and measures in DEA. J Prod Anal 11(1):5–42

Cooper WW, Seiford LM, Zhu J (2011) Data envelopment analysis: history, models, and interpretations. In: Cooper WW, Seiford LM, Zhu J (eds) Handbook on data envelopment analysis. Springer, Berlin, pp 1–39

Du J, Liang L, Chen Y, Cook WD, Zhu J (2011) A bargaining game model for measuring performance of two-stage network structures. Eur J Oper Res 210(2):390–397

Eftekhar M, Li H, Van Wassenhove LN, Webster S (2017) The role of media exposure on coordination in the humanitarian setting. Prod Oper Manag 26(5):802–816

Emrouznejad A, Yang GL (2016a) A framework for measuring global Malmquist–Luenberger productivity index with CO\(_{2}\) emissions on Chinese manufacturing industries. Energy 115:840–856

Emrouznejad A, Yang GL (2016b) \(\text{ CO }_{2}\) emissions reduction of Chinese light manufacturing industries: a novel RAM-based global Malmquist–Luenberger productivity index. Energy Policy 96:397–410

Emrouznejad A, Parker BR, Tavares G (2008) Evaluation of research in efficiency and productivity: a survey and analysis of the first 30 years of scholarly literature in DEA. Soc Econ Plan Sci 42(3):151–157

Färe R, Grosskopf S (2000) Network DEA. Soc Econ Plan Sci 34:25–49

Färe R, Grosskopf S (2004) Modelling undesirable factors in efficiency evaluation: comments. Eur J Oper Res 157(1):242–245

Färe R, Grosskopf S (2009) A comment on weak disposability in nonparametric production analysis. Am J Agric Econ 91(2):535–538

Fukuyama H, Weber WL (2010) A slacks-based inefficiency measure for a two-stage system with bad outputs. Omega 38(5):398–409

He F, Zhuang J (2016) Balancing pre-disaster preparedness and post-disaster relief. Eur J Oper Res 252(1):246–256

Huang D, Zhang R, Huo Z, Mao F, Youhao E, Zheng W (2012) An assessment of multidimensional flood vulnerability at the provincial scale in China based on the DEA method. Nat Hazards 64(2):1575–1586

Kao C, Hwang SN (2008) Efficiency decomposition in two-stage data envelopment analysis: an application to non-life insurance companies in Taiwan. Eur J Oper Res 185(1):418–429

Korhonen PJ, Luptacik M (2004) Eco-efficiency analysis of power plants: an extension of data envelopment analysis. Eur J Oper Res 154(2):437–446

Li Y, Chen Y, Liang L, Xie J (2012) DEA models for extended two-stage network structures. Omega 40(5):611–618

Li H, Chen C, Cook WD, Zhang J, Zhu J (2017a) Two-stage network DEA: who is the leader? Omega. doi:10.1016/j.omega.2016.12.009

Li WH, Li Z, Liang L, Cook WD (2017b) Evaluation of ecological systems and the recycling of undesirable outputs: an efficiency study of regions in China. Soc Econ Plan Sci. doi:10.1016/j.seps.2017.03.002

Li WH, Liang L, Cook WD (2017c) Measuring efficiency with products, by-products and parent-offspring relations: a conditional two-stage DEA model. Omega 68:95–104

Liang L, Cook WD, Zhu J (2008) DEA models for two-stage processes: game approach and efficiency decomposition. NRL 55(7):643–653

Lin TT, Lee CC, Chiu TF (2009) Application of DEA in analyzing a bank’s operating performance. Expert Syst Appl 36(5):8883–8891

Liu WB, Meng W, Li XX, Zhang DQ (2010) DEA models with undesirable inputs and outputs. Ann Oper Res 173(1):177–194

Liu W, Zhou Z, Ma C, Liu D, Shen W (2015) Two-stage DEA models with undesirable input-intermediate-outputs. Omega 56:74–87

Maghbouli M, Amirteimoori A, Kordrostami S (2014) Two-stage network structures with undesirable outputs: a DEA based approach. Measurement 48:109–118

Moreno P, Lozano S (2014) A network DEA assessment of team efficiency in the NBA. Ann Oper Res 214(1):99–124

Moreno P, Andrade GN, Meza LA, de Mello JCS (2015) Evaluation of Brazilian electricity distributors using a network DEA model with shared inputs. IEEE Latin Am Trans 13(7):2209–2216

NFPA (2016) Fire loss in the United States during 2015. http://www.nfpa.org/news-and-research/fire-statistics-and-reports/fire-statistics/fires-in-the-us/overall-fire-problem/fire-loss-in-the-united-states. Accessed on 24 Jan 2017

Peng M, Song L, Guohui L, Sen L, Heping Z (2014) Evaluation of fire protection performance of eight countries based on fire statistics: an application of data envelopment analysis. Fire Technol 50(2):349–361

Podinovski VV, Kuosmanen T (2011) Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions. Eur J Oper Res 211(3):577–585

Scheel H (2001) Undesirable outputs in efficiency valuations. Eur J Oper Res 132(2):400–410

Seiford LM, Zhu J (2002) Modeling undesirable factors in efficiency evaluation. Eur J Oper Res 142(1):16–20

Song M, An Q, Zhang W, Wang Z, Wu J (2012) Environmental efficiency evaluation based on data envelopment analysis: a review. Renew Sustain Energy Rev 16(7):4465–4469

Song M, Zhang G, Fang K, Zhang J (2016a) Regional operational and environmental performance evaluation in China: non-radial DEA methodology under natural and managerial disposability. Nat Hazards 84(1):243–265

Song M, Zheng W, Wang Z (2016b) Environmental efficiency and energy consumption of highway transportation systems in China. Int J Prod Econ 181:441–449

Song M, Peng J, Wang J, Zhao J (2017) Environmental efficiency and economic growth of China: a Ray slack-based model analysis. Eur J Oper Res. doi:10.1016/j.ejor.2017.03.073

Sueyoshi T, Goto M (2012a) Data envelopment analysis for environmental assessment: comparison between public and private ownership in petroleum industry. Eur J Oper Res 216(3):668–678

Sueyoshi T, Goto M (2012b) Weak and strong disposability vs. natural and managerial disposability in DEA environmental assessment: comparison between Japanese electric power industry and manufacturing industries. Energy Econ 34(3):686–699

Toyasaki F, Wakolbinger T (2014) Impacts of earmarked private donations for disaster fundraising. Ann Oper Res 221(1):427–447

Üstün AK (2016) Evaluating İstanbul’s disaster resilience capacity by data envelopment analysis. Nat Hazards 80(3):1603–1623

Üstün AK, Barbarosoğlu G (2015) Performance evaluation of Turkish disaster relief management system in 1999 earthquakes using data envelopment analysis. Nat Hazards 75(2):1977–1996

Wei YM, Fan Y, Lu C, Tsai HT (2004) The assessment of vulnerability to natural disasters in China by using the DEA method. Environ Impact Assess Rev 24(4):427–439

Wu J, Zhu Q, Chu J, Liang L (2015) Two-stage network structures with undesirable intermediate outputs reused: a DEA based approach. Comput Econ 46(3):455–477

Wu J, Zhu Q, Chu J, Liu H, Liang L (2016a) Measuring energy and environmental efficiency of transportation systems in China based on a parallel DEA approach. Transp Res Part D Transp Environ 48:460–472

Wu J, Zhu Q, Ji X, Chu J, Liang L (2016b) Two-stage network processes with shared resources and resources recovered from undesirable outputs. Eur J Oper Res 251(1):182–197

Yang H, Pollitt M (2010) The necessity of distinguishing weak and strong disposability among undesirable outputs in DEA: environmental performance of Chinese coal-fired power plants. Energy Policy 38(8):4440–4444

Yang F, Yuan Q, Du S, Liang L (2016) Reserving relief supplies for earthquake: a multi-attribute decision making of China Red Cross. Ann Oper Res 247(2):759–785

Yu Y, Shi Q (2014) Two-stage DEA model with additional input in the second stage and part of intermediate products as final output. Expert Syst Appl 41(15):6570–6574

Zhou P, Ang BW (2008) Linear programming models for measuring economy-wide energy efficiency performance. Energy Policy 36(8):2911–2916

Zhou P, Ang BW, Poh KL (2006) Slacks-based efficiency measures for modeling environmental performance. Ecol Econ 60(1):111–118

Zhou P, Poh KL, Ang BW (2007) A non-radial DEA approach to measuring environmental performance. Eur J Oper Res 178(1):1–9

Zhou P, Ang BW, Poh KL (2008) Measuring environmental performance under different environmental DEA technologies. Energy Econ 30(1):1–14

## Acknowledgements

The authors would like to thank the Editor of OR Spectrum and three anonymous reviewers for their insightful comments and suggestions. This research is financially supported by the Science Funds for Creative Research Groups of the National Natural Science Foundation of China (No. 71121061) and the Fund for International Cooperation and Exchange of the National Natural Science Foundation of China (No. 71110107024). This paper was finished when Feng Li was visiting the State University of New York at Buffalo with financial support from the China Scholarship Council (No. 201606340017), and Qingyuan Zhu was visiting University of Illinois at Urbana-Champaign with financial support from the China Scholarship Council (No. 201606340054).

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## Appendix

### Appendix

### Appendix A: Calculation process of model (5)

Substituting Eq. (4) into model (3), we have

By using C–C transformation (Charnes and Cooper 1962), the above model (A1) is changed into the following model (A2).

The above model is still nonlinear, by replacing \(\vartheta ^{{\prime }}\alpha =\sigma ^{{\prime }}\) we have the following linear mathematical programming,

This completes the calculation process of model (5) derived from model (3) and (4). \(\square \)

### Appendix B: Proof of Theorem 1.

**Theorem 1** *Model* (11) *will always be feasible if* \(t_d \le 1/{\left( {2R^{t}} \right) },z_{pd} \le 1/{\left( {2qR_p^z } \right) }\).

### Proof

Consider \(\hat{{w}}_i =\hat{{\delta }}=0\left( {i=1,\ldots ,m} \right) ,\hat{{\phi }}=1/{\left( {2t_d } \right) },\hat{{\nu }}_p =1/{\left( {2q\cdot z_{pd} } \right) }\) \(\left( {p=1,\ldots ,q} \right) \) and \(\hat{{\mu }}_r =R_r^y \left( {r=1,\ldots ,s} \right) \), then there would exist large enough \(\hat{{u}}^{A}\) and \(\hat{{u}}^{B}\) and according \(\left( {\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right) \) that can make the following constraints satisfied.

Further, we have \(\sum _{i=1}^m {\hat{{w}}_i x_{id} } +\hat{{\phi }}t_d +\sum _{p=1}^q {\hat{{\nu }}_p z_{pd} } =0+1/2+1/2=1\), and

Therefore, we can conclude that \(\left( {\hat{{w}}_i ,\hat{{\delta }},\hat{{\phi }},\hat{{\nu }}_p ,\hat{{\mu }}_r ,\hat{{u}}^{A},\hat{{u}}^{B},\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right) \) can make all constraints in model (11) satisfied. Therefore, it is a feasible solution to model (11). \(\square \)

### Appendix C: Proof of Theorem 2.

**Theorem 2** *The objective function of model* (11) *is unit invariant*.

### Proof

Suppose \(\left( {\hat{{w}}_i ,\hat{{\delta }},\hat{{\phi }},\hat{{\nu }}_p ,\hat{{\mu }}_r ,\hat{{u}}^{A},\hat{{u}}^{B},\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right) \) is an optimal solution to model (11), thus

Now without loss of generality, suppose that these input and output measures are multiply by a series of constants \(\left( {\beta _i ,\chi ,\pi _p ,\psi _r } \right) \), respectively. It is easy to demonstrate that \(\left( {{\hat{{w}}_i }/{\beta _i },{\hat{{\delta }}}/\chi ,{\hat{{\phi }}}/\chi ,{\hat{{\nu }}_p }/{\pi _p },{\hat{{\mu }}_r }/{\psi _r },\hat{{u}}^{A},\hat{{u}}^{B},\hat{{\zeta }}_1 ,\hat{{\zeta }}_2 } \right) \) is an optimal solution to model (11) when new data are used. Here we have

By simple algebraic calculation, we have \({\tilde{S}}_d^*=S_d^*\). This completes the proof. \(\square \)

### Appendix D: Proof of Theorem 3.

**Theorem 3** \(\mathrm{DMU}_d \left( {d=1,\ldots ,n} \right) \)
*is overall efficient if and only if both subsystems are efficient*.

### Proof

Take the situation in which the first stage dominates the overall two-stage process for example. First, if \(\mathrm{DMU}_d \left( {d=1,\ldots ,n} \right) \) is efficient, then the inefficiencies in the two stages can be minimised to be zero simultaneously. As a result, the inefficiency to each stage would also be zero. And \(E_{1d}^{1*} =E_{2d}^{1*} =1\) would be held.

Next, if two subsystems are efficient with an efficiency score of unity, namely \(E_{1d}^{1*} =E_{2d}^{1*} =1\), then \(E_d^*=E_d^{1*} +E_d^{2*} -1=1\), which indicates that the whole system is efficient. \(\square \)

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Li, F., Zhu, Q. & Zhuang, J. Analysis of fire protection efficiency in the United States: a two-stage DEA-based approach.
*OR Spectrum* **40, **23–68 (2018). https://doi.org/10.1007/s00291-017-0490-2

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### Keywords

- Data envelopment analysis
- Fire protection efficiency
- Two-stage network
- Undesirable outputs
- Managerial disposability